Questions tagged [jordan-normal-form]

This tag is for questions relating to the Jordan normal form, also known as a Jordan canonical form or JCF of a matrix which is a similar block matrix having diagonal blocks when the matrix is diagonalizable and diagonal + nilpotent blocks more generally.

The Jordan normal form of a matrix is a canonical form given by a similarity transformation. The Jordan normal form consists of diagonal blocks corresponding to its generalized eigenvectors, with blocks that are themselves constant diagonal when an eigenvalue has a basis of eigenvectors and otherwise blocks that are constant diagonal + nilpotent. For more information, see this Wikipedia article.

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If $T^k = Id$ for $k\ge 1$ then $T$ is diagonalizable

Let $V$ a finite dimension space over $\mathbb{C}$ and $T:V\to V$, a linear transformation such that $T^k = Id$ for $k\ge 1$. Prove that $T$ is diagonalizable. I'd be glad for an hint. How do I approach this question? Thanks.
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How can same number of linearly independent vectors exist in smaller power nullspaces of a matrix?

Suppose I have a $n \times n$ matrix $A$. Let the dimension of null-space of $A^k$(denoted by $\mathcal{N}_{A^k}$) be $2$. This implies there can be at most $2$ linearly independent vectors $u,v$ in $\mathcal{N}_{A^k}$. Let $\mathcal{W} =…
curryage
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Jordan Canonical Form transition matrix

I have this matrix $M$ $M = \begin{bmatrix} 1 & 1 & 1\\ 2 & 1 & -1\\ 0 & -1 & 1 \end{bmatrix}$ And I was asked to put it into Jordan Canonical Form. I did this, and obtained if $M$ = $SQS^{-1}$ Then $S =…
Arbitrationer
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Write the Jordan form of an operator

These are the properties that apply to the operator $A$. $k_A(x)=x^4(x-2)^4, d(A)=2, d(A^2)=4, d((A-2I))=2, (d((A-2I)^2)=3$ $d$ denotes the defect. $k_A$ is the characteristic polynomial. I honestly don't know where to go from here.
Luka Horvat
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Rational Canonical Form of 2x2 Matrix

I have a 2x2 matrix where I need to find the rational canonical form over the field of rational numbers and real numbers. The matrix given is $$A=\begin{pmatrix}2 &-1 \\ 1 & -1\end{pmatrix}.$$ What I've done is found the characteristic and minimal…
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Help with constructing a jordan form basis

My question is concerning Lemma 2.3 in the following paper , "On the Irreducibility of Commuting Varieties of Nilpotent Matrices " Specifically, I'm trying to understand the second half of the proof where Basili shows that we can construct a basis…
Damon
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Finding Jordan canonical basis from Jordan canonical form for 4x4 matrix

The matrix in standard basis: $$A =\begin{bmatrix}-3&1&3&3\\-10&2&9&9\\-4&0&5&4\\2&1&-3&-2\end{bmatrix}$$ characteristic polynomial is $(λ−1)^3 (λ+1)$ which means eigenvalues are $λ_1= 1$, with multiplicity $3$, and $λ_2= -1$, with multiplicity…
combat
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Solving System of Equations using Gauss/Gauss-Jordan (Matrix)

System of equations $$\begin{align*} x -2y +3z &= 2\\ 2x -3y +8z &= 7\\ 3x -4y +13z &= 8 \end{align*}$$ In a augmented matrix, $3\times 4$ $$\left(\begin{array}{crc|c} 1 &-2 &3 &2\\ 2 &-3 &8& 7 \\ 3 &-4 &13&…
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Find jordan from of matrix

I' am trying to find Jordan form of given matrix: \begin{bmatrix} 1 & 2 & 0 \\ -1 & -1 & -1 \\ 0 & 0 & 1 \end{bmatrix} So far i founD characteristic polynomial : $(1 - \lambda )[(1-\lambda)(-1-\lambda)+2]$, eingenvalue: $\lambda = 1$ and…
anna
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An orthogonal matrix in $\mathbb{R}^{3\times3}$ with real eigenvalues is diagonalizable

I know there are two non trivial (i.e. if we solve these two cases the other cases are trivial) cases: $\lambda_{1,2,3}=1$ and: $\lambda_1=1,\lambda_{1,2}=-1$ I have been trying to use generalized eigenvectors and the Jordan Canonical and the fact…
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Jordan normal form of $ A + \alpha I$

Help me please, I tried to prove, that JNF of matrix $ A + \alpha I$ is equal to matrix $ A_j + \alpha I$ where $A_j$ is JNF of A. Is it true, that $ A_j + \alpha I$ - JNF by definition? Because, it is block diagonal matrix and every block are…
GThompson
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Find the permutation matrix

Let: $$J=\begin{bmatrix} \lambda&1&0\\ 0&\lambda&1\\ 0&0&\lambda \end{bmatrix}$$ Find a permutation matrix $M$ such that $$M J M^{-1} = J^{t}$$ I know that $J$ is a Jordan form matrix, but I don't even have an idea as to how to approach the…
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If $A \in M_2(\mathbb{C})$ and $A^2 = 0$ then what are the possible forms of A?

Let $A \in M_2(\mathbb{C})$ If $A^2 = 0$ determine all of the JCF's possible. If $A^2 = 0$ determine all of the possible A's. Show that $A^2 = 0 \;\exists n \geq 2 \Leftrightarrow A^2 = 0 \;\forall n \geq 2$. For 1, the possible JCF's correspond…
kakashi10192020
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let J(A) be the Jordan form of A. and let f be some polynomial. is it true that $\det(xI-f(A))=\det(xI-f(J(A))$

I tried a couple of examples and it turned out to be true, but I couldn't prove it..
Omri Attal
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Jordan normal form options from minimal polynomial

what is the Jordan normal form options from this minimal polynomial () = $^3 − 2x^2$. 4X4 matrix I know of course that the 2 eigenvalues are 0,2.
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